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where the latter equality follows from the fact that G(K) x,r(x) has trivial Gal(K/k)-cohomology,by the pro-finite version of the Lang-Steinberg theorem.Fix a nontrivial character χ : f + → C × of the additive group of f. The compositionχ λ := χ ◦ λ : V x (f) → C ×is a character of J x which trivial on J + x , and H x,λ is the stabilizer of χ λ in H x . We consider thecompactly-induced representationπ x (λ) := ind G(k)J xχ λ ,and the intertwining algebraH x,λ = End Hx,λ(ind H x,λJ xχ λ).If ρ is a simple H x,λ -module, let χ λ,ρ denote the corresponding irreducible constitutent for H x,λ inind H x,λJ xχ λ , as in (2).Proposition 2.4. Suppose that λ ∈ ˇV x (f) is an f-rational stable functional for the action of G x onˇV x . Then the following hold.1. The representation π x (λ) has a finite direct sum decompositionπ x (λ) =⊕dim ρ · π x (λ, ρ),ρ∈Irr(H x,λ )where π x (λ, ρ) := ind G(k)H x,λχ λ,ρ is an irreducible supercuspidal representation of G(k), foreach ρ ∈ Irr(H x,λ ).2. If ρ and ρ ′ are inequivalent simple modules for H x,λ then π x (λ, ρ) and π x (λ, ρ ′ ) are inequivalentrepresentations of G(k).3. The formal degree of π x (λ, ρ) with respect to a Haar measure µ on G(k) is given bydeg µ (π x (λ, ρ)) = dim χ λ,ρ|A x,λ |·1µ(J x ) .Proof. By Lemma 2.2, it suffices to show that if g ∈ G(k) andχ λ = χ g λon J x ∩ J g x, (7)12